Fourier instantaneous estimators and the Epps effect
FFourier instantaneous estimators and the Epps e ff ect Patrick Chang a a Department of Statistical Science, University of Cape Town, Rondebosch 7700, South Africa
Abstract
We compare the Malliavin-Mancino and Cuchiero-Teichmann Fourier instantaneous estimators to investigate the impact of the Eppse ff ect arising from asynchrony in the instantaneous estimates. We demonstrate the instantaneous Epps e ff ect under a simulationsetting and provide a simple method to ameliorate the e ff ect. We find that using the previous tick interpolation in the Cuchiero-Teichmann estimator results in unstable estimates when dealing with asynchrony, while the ability to bypass the time domain withthe Malliavin-Mancino estimator allows it to produce stable estimates and is therefore better suited for ultra-high frequency finance.An empirical analysis using Trade and Quote data from the Johannesburg Stock Exchange illustrates the instantaneous Epps e ff ectand how the intraday correlation dynamics can vary between days for the same equity pair. Keywords:
Malliavin-Mancino estimator, Cuchiero-Teichmann estimator, Instantaneous estimates, Epps e ff ect
1. Introduction
The study of the integrated volatility / co-volatility is a wellexplored field with a plethora of estimators suited for variouspurposes. To name a few, we have the classical Realised Volatil-ity (RV) for the continuous-time Itˆo semi-martingale. Jumprobust extensions such as the Bi- and Multi-Power variationsstudied by Barndor ff -Nielsen and Shephard (2004); Barndor ff -Nielsen et al. (2006a,b), which has later been generalised by re-placing the power function with di ff erent specifications (See Ja-cod (2008); Todorov and Tauchen (2012); Podolskij and Vetter(2009)). Extensions to deal with asynchrony such as the cumu-lative estimator proposed by Hayashi and Yoshida (2005), andFourier estimators such as that proposed by Malliavin and Man-cino (2002, 2009). On the other hand, the study of the instan-taneous volatility / co-volatility is a relatively new field. Most ofthe instantaneous estimators rely on the numerical derivative ofthe integrated covariance (See Alvarez et al. (2012); Bandi andRen (2018); Mykland and Zhang (2008)), which correspond tothe local realised volatility estimator . Due to the di ff erentiation,this can lead to strong numerical instabilities.Instantaneous estimators based on Fourier transforms presentseveral advantages over di ff erentiation based local RV estima-tors. First, it relies on the integration of the time series ratherthan the di ff erentiation which makes it numerically stable. Sec-ond, the reconstruction of the instantaneous covariance relies onharmonics. Therefore, the degree of smoothness is determinedby the cutting frequency M for the reconstruction. Finally, themethods provide global estimation of the spot volatility. Mean-ing the volatilities are estimated with similar accuracy at anytime t within the interior of the domain (Mancino et al., 2017).The spot volatility has a range of applications. The inte-grated stochastic volatility of volatility can be estimated by us- Email address:
[email protected] (Patrick Chang) ing power variation estimators on the reconstructed instanta-neous volatility path (Cuchiero and Teichmann, 2015), or ap-proaches that rely only on integrated quantities (Sanfelici et al.,2015). Another parameter which can also be obtained from thespot estimates is the integrated leverage investigated in Curatoand Sanfelici (2015). Now both the instantaneous volatility ofvolatility and spot leverage can be obtained as pointed out inMancino et al. (2017). Other applications include predicting theintraday Value at Risk (VaR) Mancino and Recchioni (2015),calibrating agent based models such as the model in Tedeschiet al. (2012), in the medical field to approximate heart rate vari-ability Mancino and Recchioni (2015), or in the environmentalfield to studying ecological systems and changes of biodiversityvariability in its variation in space and time (Costa et al., 2018;Ferrauto et al., 2013).Here we are interested in comparing the Malliavin-Mancino(Malliavin and Mancino, 2009) and the Cuchiero-Teichmann(Cuchiero and Teichmann, 2015) Fourier instantaneous estima-tors and understanding the impact of their cutting frequenciesto gain insight into the impact of the Epps e ff ect (Epps, 1979)on the instantaneous estimators. The Epps e ff ect is the decayof correlations as the sampling intervals decrease and there isan abundance of literature investigating the causes (See Ren`o(2003); T´oth and Kert´esz (2007, 2009); M¨unnix et al. (2010);Mastromatteo et al. (2011)) and corrections (See M¨unnix et al.(2011); Hayashi and Yoshida (2005); Zhang (2010); Buccheriet al. (2019); Chang et al. (2020c)), but all in the case of theintegrated correlation. To the best of my knowledge, apart fromthe work by Mattiussi and Iori (2010), the impact on the in-stantaneous correlation caused by the Epps e ff ect has not beenexplored. This work aims to remedy this by serving as a pre-liminary investigation into the instantaneous Epps e ff ect as newresults in applied harmonic analysis (Guariglia and Silvestrov,2016; Guariglia, 2018).To this end, we compare the two estimators for various Preprint submitted to arXiv September 28, 2020 a r X i v : . [ q -f i n . S T ] S e p tochastic models with and without jumps under synchronousand asynchronous observations to gain insight into the di ff er-ences between the estimators. We then investigate the impactof the reconstruction frequency M in the estimators, and moreimportantly, we investigate the impact of the time-scale ∆ t un-der asynchronous observations to study the instantaneous Eppse ff ect under a simulation setting. We provide an ad hoc ap-proach to deal with asynchrony based on the understanding ofthe Epps e ff ect arising from asynchrony and demonstrate thatthe e ff ect can be ameliorated for the instantaneous estimator.Finally, we demonstrate the instantaneous Epps e ff ect usingTrade and Quote data from two banking equities in the Johan-nesburg Stock Exchange and compare the two estimators aftercorrecting for the Epps e ff ect.The paper is organised as follows: Section 2 introduces thetwo Fourier estimators. Section 3 compares the volatility andco-volatility estimates of the two estimators for the variousStochastic models. Section 4 investigates the impact of the cut-ting frequencies, the time-scale under asynchrony and demon-strates how to deal with the Epps e ff ect arising from asyn-chrony. Section 5 provides the empirical demonstration. Fi-nally, Section 6 concludes.
2. Fourier instantaneous estimators
Malliavin and Mancino (2009) provided a non-parametricFourier estimator that is constructed in the frequency domain. Itexpresses the Fourier coe ffi cients of the volatility process Σ i j ( t )using the Fourier coe ffi cients of the price process X it = ln( P it )where P it is a generic asset price at time t . By re-scaling thetrading times from [0 , T ] to [0 , π ] and using Bohr convolutionproduct (See Theorem 2.1 of Malliavin and Mancino (2009))we have for all k ∈ Z : F ( Σ i j )( k ) = lim N →∞ π N + (cid:88) | s |≤ N F ( dX i )( s ) F ( dX j )( k − s ) . (2.1)Here F ( ∗ )( (cid:63) ) is the (cid:63) th Fourier coe ffi cient of the ∗ process (Ra-gusa and Shakhmurov, 2018; Ragusa, 2012). Using the previ-ous tick interpolation to avoid a downward bias in the estimator(Barucci and Ren`o, 2002) and a simple function approximationfor the Fourier coe ffi cients, we get: F ( dX i )( k ) ≈ π n i − (cid:88) h = exp( − i kt ih ) δ i ( I h ) , F ( dX j )( k ) ≈ π n j − (cid:88) (cid:96) = exp( − i kt j (cid:96) ) δ j ( I (cid:96) ) , (2.2)where ( t ih ) h = ,..., n i and ( t j (cid:96) ) (cid:96) = ,..., n j are the observation times. Theprice fluctuations are δ i ( I h ) = X it ih + − X it ih and δ j ( I (cid:96) ) = X jt j (cid:96) + − X jt j (cid:96) for asset i and j respectively. Here i ∈ C in the exponential defining the Fourier transform is such thati = √− i for the asset. The Fourier coe ffi cients of the volatility process is given as: α k (cid:16) Σ i jn i , n j , N (cid:17) = π N + (cid:88) | s |≤ N F ( dX i )( s ) F ( dX j )( k − s ) , (2.3)for k ∈ {− M , ..., M } . Therefore, the instantaneous volatility / co-volatility can be reconstructed using eq. (2.3) as:ˆ Σ i jn i , n i , N , M ( t ) = (cid:88) | k |≤ M (cid:32) − | k | M (cid:33) e i tk α k (cid:16) Σ i jn i , n j , N (cid:17) . (2.4)Notice that there are two parameters N and M that require tun-ing in eq. (2.4). Here N dictates how many Fourier modes areused in estimating the Fourier coe ffi cients of the volatility. Thiscontrols the level of averaging which has a direct implication onthe time-scale of investigation (See Chang et al. (2020b)). Thesecond parameter M is the reconstruction frequency. It deter-mines how many Fourier coe ffi cients are used in approximatingthe spot volatility. Section 4 will investigate the impact of N and M in greater detail. Moreover, notice that n i and n j neednot be the same. This is the main feature behind the Malliavin-Mancino Fourier estimator. The convolution is performed inthe frequency domain, allowing one to bypass the issue of asyn-chrony in the time domain. This means we do not need to syn-chronise the data beforehand by means of imputation.The implementation of the instantaneous estimator requiresthe evaluation of {− M − N , ..., N + M } Fourier coe ffi cients for as-set i and j which can be computationally expensive. Therefore,the implementation relies on non-uniform fast Fourier trans-forms to computationally speed up the evaluation of eq. (2.2)(See Chang et al. (2020b)). Cuchiero and Teichmann (2015) provide an extension basedon the Malliavin-Mancino Fourier estimator to account for thepresence of jumps. This is achieved by modifying jump robustestimators of integrated RVs considered by Barndor ff -Nielsenet al. (2006a,b); Jacod (2008); Podolskij and Vetter (2009);Todorov and Tauchen (2012) in order to obtain estimates forthe Fourier coe ffi cients of the realised path of the instantaneousvolatility.The Cuchiero-Teichmann spot volatility is estimated throughthree steps. First, we need an estimator for the Fourier coe ffi -cients of ρ g ( Σ ) from the discrete price observations X t where ρ g ( · ) is a continuous invertible function. Meaning we need anestimator for: F (cid:16) ρ g ( Σ ) (cid:17) ( k ) = T (cid:90) T ρ g ( Σ ( t )) e − i π T kt . (2.5)Cuchiero and Teichmann (2015) show that the estimator ofeq. (2.5) taking the form: V ( X , g , k ) nT = n (cid:98) nT (cid:99) (cid:88) h = e − i π T kt nh − g (cid:16) √ n ∆ nh X (cid:17) , (2.6)converges to the required Fourier coe ffi cients in eq. (2.5) (SeeTheorem 3.4 of Cuchiero and Teichmann (2015)). Here 1 / n = t is the discretisation interval, ∆ nh X = X t nh − X t nh − , and the timegrid for the observations of X t in [0 , T ] are equal and equidis-tant, i.e. t nh = h / n , h = , ..., (cid:98) nT (cid:99) .Second, we can apply the Fourier-Fej´er inversion to recon-struct the path of: (cid:92) ρ g ( Σ ( t )) n , M = T (cid:88) | k |≤ M (cid:32) − | k | M (cid:33) e i π T kt V ( X , g , k ) nT . (2.7)Finally, the spot volatility can be obtained by inverting eq. (2.7)yielding: ˆ Σ n , M ( t ) = ρ − g (cid:16) (cid:92) ρ g ( Σ ( t )) n , M (cid:17) . (2.8)Up till now in Section 2.2, the use of asset indices i and j have been avoided. This is because the Cuchiero-Teichmannspot volatility estimator does not share the property of theMalliavin-Mancino Fourier estimator where the Fourier coe ffi -cients of the volatility are constructed in the frequency domainvia a convolution with the Fourier coe ffi cients of the price pro-cess (See eq. (2.3)). The Fourier coe ffi cients of the volatility inthe Cuchiero-Teichmann estimator is obtained by adapting inte-grated RVs; meaning direct price observations are used, ratherthan their Fourier coe ffi cients (See eq. (2.6)). This means thatthe reconstruction of ˆ Σ n , M ( t ) requires the use of the polarisationidentity. Therefore, the Cuchiero-Teichmann estimator does nothave a method to over come the issue of asynchrony.Here, we use Todorov and Tauchen (2012) specification ofthe function g ( x ) = cos( x ), therefore ρ g ( Σ ( t )) = e − Σ ( t ) / . Nowby using the polarisation identity, we have:ˆ Σ iin , M ( t ) = − (cid:16) (cid:92) ρ g ii ( Σ ( t )) n , M (cid:17) , i ∈ { , } , ˆ Σ n , M ( t ) = (cid:16) − (cid:16) (cid:92) ρ g ( Σ ( t )) n , M (cid:17) − ˆ Σ n , M ( t ) − ˆ Σ n , M ( t ) (cid:17) . (2.9)The Cuchiero-Teichmann estimator has a tuning parameter M which is the reconstruction frequency of the spot volatil-ity. There is no parameter N controlling the level of averag-ing. Therefore, the estimator relies on the discretisation interval ∆ t = / n to investigate various time-scales.
3. Comparison
We compare the Malliavin-Mancino (MM) and Cuchiero-Teichmann (CT) spot volatility estimator under the presenceof jumps and no jumps for the synchronous and asynchronouscase. We consider two types of volatility models: constantvolatility and stochastic volatility. The stochastic models con-sidered are the Geometric Brownian Motion (GBM) for con-stant volatility with no jumps; the Merton Model for constantvolatility with jumps; the Heston Model for stochastic volatilitywith no jumps; and the Bates-type model for stochastic volatil-ity with jumps. The parameters are given for the period [0 , T ]where T = Note that δ i ( I h ) and ∆ nh X are both price fluctuations. Separate notation isused highlight that the observation times in δ i ( I h ) need not be equidistant, while ∆ nh X must be strictly equidistant. Geometric Brownian Motion
The bivariate Geometric Brownian Motion (GBM) satisfiesthe following system of SDEs dP it P it = µ i dt + σ i dW i ( t ) , i = , , (3.1)where W i are Brownian motions with Corr( dW , dW ) = ρ .The parameters used in the simulation are given in Table 1.Model Parameter ValuesGBM / Merton ( µ , µ ) (0.01, 0.01)( σ , σ ) (0.1, 0.2) ρ a , a ) (-0.005, -0.003)( b , b ) (0.015, 0.02)( λ , λ ) (100, 100) Table 1: Parameters used in the simulation for the Geometric Brownian Motionand the Merton Model.
Merton Model
The bivariate Merton model satisfies the following system ofSDEs: dP it P it − = µ i dt + σ i dW it + dJ it , i = , . (3.2)Here the correlation is Corr( dW , dW ) = ρ and the intervals J i are independent of the W i with piece-wise constant paths(Glasserman, 2004). J is defined as: J it = N ( t ) (cid:88) j = ( Y j − , (3.3)where N ( t ) is a Poisson process with rate λ i , Y j ∼ LN ( a , b )i.i.d and also independent of N ( t ). The parameters used in thesimulation are given in Table 1.
Heston Model
The stochastic volatility models are simulated such that theentire volatility matrix is stochastic. Concretely, the bi-variateHeston model takes the form: X t = X + (cid:90) t − Σ diag ( s ) ds + (cid:90) t (cid:112) Σ ( s ) dZ s , Σ ( t ) = Σ (0) + (cid:90) t (cid:16) b + M Σ ( t ) + Σ ( t ) M (cid:62) (cid:17) dt + (cid:112) Σ ( t ) dB t H + HdB (cid:62) t (cid:112) Σ ( t ) , (3.4)where M and H are invertible matrices, Σ (0) a 2-dimensionalpositive semi-definite symmetric matrix S + , b − H ∈ S + , and Z is a 2-dimensional Brownian motion correlated with the 2x2matrix of Brownian motions B such that Z = (cid:112) − ρ (cid:62) ρ W + B ρ , The t − on the LHS of (3.2) is used to indicate the C`adl`ag nature of theprocess near jumps. ρ ∈ [ − , such that ρ (cid:62) ρ ≤ W is a 2-dimensionalBrownian motion independent of B . The parameters for thesimulation are given in Table 2.Model Parameter ValuesHeston / Bates ( X , X ) (4.6, 4.6) (cid:32) Σ (0) Σ (0) Σ (0) Σ (0) (cid:33) (cid:32) . − . − .
036 0 . (cid:33) M (cid:32) − . − . − . − (cid:33) α = H (cid:32) . . .
06 0 . (cid:33) b . αρ (-0.3, -0.5)Bates ( λ X , λ X ) (100, 100)( a , a ) (-0.005, -0.003)( b , b ) (0.015, 0.02) λ Σ θ Table 2: Parameters used in the simulation for the Heston and the Bates-typeModel. The parameters are borrowed from Cuchiero and Teichmann (2015).
Bates model
Here we consider a Bates-type model (henceforth referred toas the Bates model for simplicity) where jumps occur in the log-price X t and in the volatility process Σ ( t ). The 2-dimensionalBates model takes the form: X t = X + (cid:90) t b s ds + (cid:90) t (cid:112) Σ ( s − ) dZ s + (cid:90) t (cid:90) R ξµ X ( d ξ, ds ) , Σ ( t ) = Σ (0) + (cid:90) t (cid:16) b + M Σ ( t ) + Σ ( t ) M (cid:62) (cid:17) dt + (cid:112) Σ ( t ) dB t H + HdB (cid:62) t (cid:112) Σ ( t ) + (cid:90) t (cid:90) R ξµ Σ ( d ξ, ds ) . (3.5)The Brownian motions Z and B , and the parameters b , H and M are defined as before in the Heston model. Here µ X ( d ξ, dt ) isthe random measure associated with the jumps of X . The jumpsare Gaussian with mean a i , standard deviation b i , and the rate ofjumps is λ Xi . Here µ Σ ( d ξ, dt ) is the random measure associatedwith the jumps of Σ . The jumps are exponential with parameter θ and the rate of jumps is λ Σ . Lastly, the drift of X is givenby b s , i = − Σ ii ( s ) − λ Xi (cid:16) e a i − b i − (cid:17) . The parameters for thesimulation are given in Table 2. Let us consider the impact of jumps against no jumps forthe two spot volatility estimators when the observations are ob-served on an equidistant synchronous grid ( n = n = n ). Thenumber of grid points is set to n = ,
800 which corresponds The jumps in the volatility process only happen for Σ ( t ) as in Cuchieroand Teichmann (2015). to 1-second data for a trading day of 8 hours. The four afore-mentioned stochastic models are simulated using the parame-ters given in the tables.Figure 1 compares the true underlying volatility (black andlight-blue lines) against the Malliavin-Mancino (blue lines) andCuchiero-Teichmann (red dashes) spot volatility estimates. Therows of the figures are: GBM, Merton, Heston, and Bates modelfrom first to last. The columns of the figures are: Σ ( t ), Σ ( t ),and Σ ( t ) from first to last. The cutting frequencies used are N as the Nyquist frequency for the Malliavin-Mancino estimator,and M =
100 for both the Malliavin-Mancino and Cuchiero-Teichmann estimator. For the GBM and Heston model, we seethat both the Malliavin-Mancino and Cuchiero-Teichmann es-timators recover the entire volatility matrix with high fidelity.For the Merton and Bates model, we see that the volatility( Σ ii ( t ), i = ,
2) estimates using the Malliavin-Mancino esti-mator presents regions with spikes in volatility when there arejumps. On the other hand, the Cuchiero-Teichmann estima-tor does not present large spikes in volatility because it is ro-bust to jumps. However, the e ff ect of jumps can still be seenin the Cuchiero-Teichmann estimator as the volatility becomesslightly higher than in the case with no jumps. Interestingly, theco-volatility ( Σ ( t )) estimates are not severely altered by jumpsfor both the Malliavin-Mancino and Cuchiero-Teichmann esti-mators. Jumps result in a negative bias in the integrated correla-tion estimate (Chang et al., 2019), here we see that the negativebias is a result of volatility spikes which results in a larger nor-malisation factor for the correlation and ultimately leads to adownward bias. Let us investigate the additional impact of asynchrony on topof jumps against no jumps for the spot volatility estimators.Asynchrony is introduced using an arrival time representation,where the inter-arrival time between trades follow an exponen-tial distribution with parameter λ (also known as Poissoniansampling). Thus the mean inter-arrival time is given as 1 /λ .The Cuchiero-Teichmann estimator does not have the abilityto deal with asynchronous observations, so we need a method tosynchronise the data at an appropriate time-scale. We will usethe previous tick interpolation to impute the asynchronous ob-servations onto a new uniform grid. To this end, let U i = { t ik } k ∈ Z ,be the set of asynchronous arrival times observed between[0 , T ] for asset i . The synchronised process is then given by˜ X it = X i γ i ( t ) , where γ i ( t ) = max { t ik : t ik ≤ t } for t ∈ [0 , T ]. The re-sulting synchronised process is piece-wise constant with jumpsat t ik ∈ U i . The new uniform grid has width ∆ t , which will bethe time-scale of investigation.The Malliavin-Mancino estimator deals with asynchrony byperforming the operations in the frequency domain where asyn-chrony is not an issue anymore. Moreover, the estimator hasan implicit method to investigate various time-scales throughan appropriate choice of N . This is discussed in Chang et al.(2020b). The link between ∆ t and N is given by: N = (cid:36) (cid:18) T ∆ t − (cid:19)(cid:37) , (3.6)4 .00 0.25 0.50 0.75 1.000.050.100.150.200.250.300.35 (a) GBM Σ ( t ), N = Nyq., M = (b) GBM Σ ( t ), N = Nyq., M = (c) GBM Σ ( t ), N = Nyq., M = (d) Merton Model Σ ( t ), N = Nyq., M = (e) Merton Model Σ ( t ), N = Nyq., M = (f) Merton Model Σ ( t ), N = Nyq., M = (g) Heston Model Σ ( t ), N = Nyq., M = (h) Heston Model Σ ( t ), N = Nyq., M = (i) Heston Model Σ ( t ), N = Nyq., M = (j) Bates Model Σ ( t ), N = Nyq., M = (k) Bates Model Σ ( t ), N = Nyq., M = (l) Bates Model Σ ( t ), N = Nyq., M = / co-volatility estimator for stochastic models with constant volatility against stochastic volatility, and jumps against no jumps. The models are simulatedwith n = ,
800 synchronous grid points. The spot estimates are compared against the true underlying instantaneous volatility matrix (black and light-blue lineswith label “True”). We see that for the models with no jumps, both estimators recover the instantaneous volatility matrix with high accuracy. For the models withjumps, the Malliavin-Mancino volatility estimates have spikes in volatility caused by jumps while the Cuchiero-Teichmann volatility estimates are not as severelya ff ected. In the case of the co-volatility estimates, both estimators are not a ff ected by jumps and recover the underlying co-volatility. .00 0.25 0.50 0.75 1.000.00.10.20.30.4 (a) GBM Σ ( t ) (b) GBM Σ ( t ) (c) GBM Σ ( t ) (d) Merton Model Σ ( t ) (e) Merton Model Σ ( t ) (f) Merton Model Σ ( t ) (g) Heston Model Σ ( t ) (h) Heston Model Σ ( t ) (i) Heston Model Σ ( t ) (j) Bates Model Σ ( t ) (k) Bates Model Σ ( t ) (l) Bates Model Σ ( t )Figure 2: Here we compare the Malliavin-Mancino (blue line with label “Estimated MM”) and Cuchiero-Teichmann (red dashes with label “Estimated CT”)spot volatility / co-volatility estimator for stochastic models with constant volatility against stochastic volatility, and jumps against no jumps under the influence ofasynchrony. Asynchrony is introduced by sampling each synchronous grid with n = ,
800 grid points using an exponential inter-arrival with mean 30, yielding n i ≈ n /λ i . A time-scale of ∆ t = ff ectof jumps increasing the volatility of the estimates are further confounded with the e ff ects of asynchrony in the Malliavin-Mancino estimator. Both estimators haveco-volatility estimates around zero due to the Epps e ff ect arising from asynchrony. T is the entire interval of investigation. T should be mea-sured in the same units as ∆ t , which is seconds in our case. Thismeans T = ,
800 seconds for an 8 hour trading day. Note thatthe conversion may not be exact because N is an integer.We simulate n = ,
800 synchronous grid points for theabove processes. These are then each sampled with an expo-nential with a mean of 1 /λ i =
30 seconds. Thus n i ≈ T /λ i asynchronous observations for the Malliavin-Mancino estima-tor. We pick N such that ∆ t = X it is constructed using the previous tick interpolation with ∆ t = M is chosen to again be100 so that comparisons can be made with Figure 1.Figure 2 compares the true underlying volatility (black andlight-blue lines) against the Malliavin-Mancino (blue lines) andCuchiero-Teichmann (red dashes) spot volatility estimates un-der the presence of asynchrony. The rows of the figures are:GBM, Merton, Heston, and Bates model from first to last. Thecolumns of the figures are: Σ ( t ), Σ ( t ), and Σ ( t ) from first tolast. Before in Figure 1, the Cuchiero-Teichmann could recoverthe entire volatility matrix with high fidelity for the models withjumps and no jumps. Here we now see that the volatility ( Σ ii ( t ), i = ,
2) is under-estimated. This is because of the previous tickinterpolation, since ∆ t = ff ectis enhanced where there are jumps. The reason behind the de-cay of the covariance estimate is due to the Epps e ff ect, which isthe decay of correlations as the sampling interval ∆ t decreases(See Epps (1979)).We will investigate the impact of the Epps e ff ect arising fromasynchrony for the instantaneous correlation in Sections 4.2and 4.3. Here we have demonstrated the e ff ect of small ∆ t (large N ) on the individual volatility components. The resultsare consistent with the case of the integrated volatility. Mancinoet al. (2017) performed a bias-MSE analysis for the Malliavin-Mancino integrated covariance estimates under the conditionsof asynchrony. They found that the integrated volatility presentslittle bias for any choice of N , while larger N results in a largerbias for the case of the integrated co-volatility which we havedemonstrated in the case of the instantaneous volatility.
4. Cutting frequencies
In Section 3, the cutting frequencies were chosen somewhatarbitrarily. However, the impact of the cutting frequencies N and M play a pivotal role in the Fourier spot volatility esti-mates. For the Malliavin-Mancino estimator, N and M a ff ects the asymptotic properties and convergence rates of the esti-mators (Mancino and Recchioni, 2015; Mancino et al., 2017;Chen, 2019). The level of averaging N determines the time-scale for which to estimate eq. (2.3), which controls for the im-pact of the Epps e ff ect caused by asynchrony. The reconstruc-tion frequency M determines the accuracy of the approxima-tion, the time-scale for the reconstruction of the volatility ma-trix, and also a ff ects the volatility of volatility estimation eitherthrough the path constructed (Cuchiero and Teichmann, 2015),or by the number of Fourier modes used in the estimation (San-felici et al., 2015).Several choices for N and M have been put forward in theliterature. Mancino et al. (2017) suggest to use N = n / M = π √ n log n for the synchronous case; N = . n / and M = π √ n / log n / for a special case of asynchronywhere n = n i = n j and one of the process is observed onnon-equidistant grid points and the other on a synchronous grid.Their choice comes from satisfying asymptotic properties andsimulation experiments. Chen (2019) suggests N ≤ (cid:98) n / (cid:99) − M for the synchronous case, and N = o ( n / ) for the asynchronouscase where n = min i n i . Chen (2019) places the additionalrestriction that N + M must be less than equal the Nyquistfrequency (of the price process). This is because ultimatelyeqs. (2.3) and (2.4) are estimated from the Fourier coe ffi cientsof the price process eq. (2.2). Thus the condition is placed toavoid aliasing. Chen (2019) motivates these choices throughconvergence and asymptotic properties. Mattiussi and Iori(2010) use the Nyquist frequencies N = n / M = N / δ which al-lows them to tune the process to the desired time-scale. Theyadopt a di ff erent approach by choosing δ to minimise the MSEin an attempt to determine if there exists an optimal time-scalesfor reconstructing the instantaneous volatility matrix. Here wevisualise the impact of M and N in order to design a pragmaticapproach to pick N and M . The cutting frequency M is the number of harmonics usedin the reconstruction of the spot volatility. Let us consider anexperiment to demonstrate how this works. We simulate n = ,
800 grid points using a Heston model with parameters inTable 2. First we consider the synchronous case and visualisethe impact of various choices of M has on the instantaneouscorrelation estimate ˆ ρ ∆ t ( t ) in Figure 3.The first row of Figure 3 are the Malliavin-Mancino spotestimates for M ranging from 1 to 100 plotted as a surfaceplot and a contour plot; analogous to the first row, the secondrow are the Cuchiero-Teichmann spot estimates. The last rowis the Malliavin-Mancino (blue line) and Cuchiero-Teichmann(red dashes) spot estimates compared against the ground truth(light-blue line) for M = N is chosen to be the Nyquist frequency. From thesurface and contour plots, we see how the harmonics build upto achieve a better approximation by allowing the estimates toreach higher peaks and lower troughs. Moreover, we see that7hen M is too small the spot estimates cannot recover the finerdetails. However, we note that when M is too large the spot esti-mates presents a rapid zigzagging behaviour. How one achievesthis subtle balance is still an open question relating to if thereexist an optimal time-scale to reconstruct the spot volatility es-timates (Mancino et al., 2017). It must be noted that the choiceof M = N / δ in the modified Fej´er kernel.Based on the preliminary investigation here, having a onesize fit all choice for M does not seem to be feasible. This isbecause a good choice of M depends on the composition of thetrue spot parameter of interest. For example, the instantaneousstochastic correlation in the Heston model takes on a more com-plex form and therefore requires more harmonics M to providea good approximation. If we use the synchronous choice of M from Mancino et al. (2017) we obtain M =
34 for this experi-ment, which from Figure 3 does not provide a good approxima-tion. On the other hand, if the instantaneous correlation takeson a simple shape such as Figure 5 then few harmonics are al-ready enough to obtain a good approximation. Therefore, ad-ditional harmonics adds redundant information which results inan unsatisfactory approximation (Mattiussi and Iori, 2010). Theconundrum presented here is based on the synchronous case,before adding the additional complication of asynchrony. Withthe added impact of asynchrony, one needs to first pick an ap-propriate N to avoid the Epps e ff ect before deciding on an ap-propriate choice of M . The cutting frequency N is the number of Fourier coe ffi -cients of the price process used to estimate the Fourier coef-ficients of the volatility. For the synchronous case, Mancinoet al. (2017) have argued that without the presence of asyn-chrony or microstructure noise the Nyquist frequency for N isthe best choice. Here we want to investigate the impact that theEpps e ff ect arising from asynchrony has on the instantaneouscorrelation through the implied time-scale ∆ t by choice of N ,analogous to the case of integrated correlation in Chang et al.(2020b); Ren`o (2003). In the case of the Cuchiero-Teichmannestimator, we will use the previous tick interpolation to syn-chronise the asynchronous samples to a particular time-scale.To this end, we simulate n = ,
800 grid points using a Hestonmodel with parameters in Table 2. The synchronous process isthen each sampled with an exponential inter-arrival with a meanof 30 seconds giving n i ≈ n /λ i .Figure 4 investigates ∆ t ranging from 1 to 100 seconds forthree cases of M . Concretely, the columns of the figures are M = ,
20 and 50 respectively while the first two rows arethe Malliavin-Mancino estimates and the last two rows are theCuchiero-Teichmann estimates. There are three things to no-tice in these figures. First, we see that when ∆ t is small thecorrelation remains around zero; while the correlations starts toemerge as ∆ t increases for the various choices of M . This is ademonstration of the instantaneous Epps e ff ect. Second, M isseverely a ff ected by asynchrony. The instantaneous correlationpresents the zigzagging behaviour for much smaller choices of M relative to the synchronous case. For example, the choiceof M =
100 in Figure 3e resulted in a relatively smooth plot;while for the asynchronous case M =
50 in Figures 4c and 4ipresents more rapid fluctuations. This is especially true for theCuchiero-Teichmann estimates which leads to the third point.Here the Cuchiero-Teichmann estimates do not only presentrapid fluctuations for fixed ∆ t and varying time t (caused bylarger M ), it also presents rapid fluctuations for fixed t andvarying ∆ t . This means the Cuchiero-Teichmann estimates arehighly unstable under the presence of asynchrony for di ff erentvalues of ∆ t . This can be seen by the horizontal black markson the contour plots. These black marks are caused by suddenchanges in values and the horizontal nature means that for aspecific time t in eq. (2.8), the estimates can take on very di ff er-ent values for similar values of ∆ t . This is di ff erent to the blackmarks of the Malliavin-Mancino estimates in Figure 4f wherethe marks run vertically down the contour plot. These changesin estimates are from large M rather than the instability fromdi ff erent ∆ t values. This means that for a fixed point in time t of eq. (2.4), various values of ∆ t do not cause sudden changesin the estimates. This provides an interesting insight into thetwo methods dealing with asynchrony. Through bypassing thetime-domain the Malliavin-Mancino estimator is able to pro-duce stable estimates while the previous tick interpolation pro-duces highly unstable estimates for various ∆ t . This is due tothe fact that under asynchrony, the Malliavin-Mancino uses allthe available observations which are fixed. In the case of usingthe previous tick interpolation, the synchronised sample pathcan change under various choices of ∆ t . These changes in thesynchronised price paths are more apparent for larger ∆ t whichis where the instabilities are occurring in Figures 4j to 4l. Thisis the source causing the instability. The spot estimates providean interesting perspective on this because this is not easily seenin the integrated estimates as it gets hidden away in the averag-ing. Chen (2019) provided a break down for the impact of asyn-chrony. There is a trade-o ff between the rate of convergenceand the bias caused by asynchrony which he calls the “curseof asynchrony”. Moreover, he provides a su ffi cient (not neces-sary) condition to ameliorate the e ff ect. Here we look at thistrade-o ff through the lens of the Epps e ff ect. This downwardbias in the Epps e ff ect is caused by the fact that the underly-ing co-variation is extracted when the asynchronous observa-tions overlap (M¨unnix et al., 2011). There are several methodsto correct for this in the case of integrated covariances suchas using the Hayashi-Yoshida estimator (Hayashi and Yoshida,2005), directly accounting for the non-overlapping e ff ects at aparticular time-scale (Chang et al., 2020c; M¨unnix et al., 2011),or simply investigating larger time-scales (Ren`o, 2003). In thecase of the instantaneous estimates, it is not clear how the firsttwo correction methods can be applied. Therefore, the mostfeasible approach is to follow Ren`o (2003) and find a small N which corrects for the Epps e ff ect. With this in mind, the choiceof N should not be chosen as a one size fits all choice. This isbecause di ff erent types of asynchrony and di ff erent levels of8 (a) MM Surface plot, N = Nyquist (b) MM Contour plot, N = Nyquist
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 - 1.00- 0.75- 0.50- 0.2500.250.500.751.00 (c) CT Surface plot, N = Nyquist (d) CT Contour plot, N = Nyquist (e) Synchronous, N = Nyquist, M = M ranging from 1 to 100 on the instantaneous correlation for the synchronous case of a Heston model with parametersin Table 2. We simulate n = ,
800 synchronous grid points. The first row of figures is the Malliavin-Mancino estimates visualised using a surface and contourplot; the second row is that of the Cuchiero-Teichmann estimates. The figure in the last row is a comparison between the Malliavin-Mancino (blue line with label“Estimated MM”) and the Cuchiero-Teichmann (red dashes with label “Estimated CT”) estimates against the true instantaneous correlation (light-blue line withlabel “True”) for M = N . We see that as M increases the additional harmonics allow us to achieve a betterapproximation. (a) MM, M =
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (b) MM, M =
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (c) MM, M = M =
10 (e) MM, M =
20 (f) MM, M =
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (g) CT, M =
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (h) CT, M =
20 40 60 80 1000.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (i) CT, M = M =
10 (k) CT, M =
20 (l) CT, M = ∆ t ranging from 1 to 100 on the instantaneous correlation of a Heston model with parameters in Table 2 under thepresence of asynchrony. We simulate n = ,
800 synchronous grid points which is then sampled with an exponential inter-arrival with a mean of 30 seconds,resulting in n i ≈ n /λ i . The three columns of the figures are M = ,
20 and 50 while the first two rows are the Malliavin-Mancino estimates visualised as surfaceand contour plots; likewise, the last two rows are that of the Cuchiero-Teichmann estimates. The time-scale ∆ t is controlled using eq. (3.6) for the Malliavin-Mancino estimates while the time-scale for the Cuchiero-Teichmann estimates are controlled using the previous tick interpolation. We see the visualisation of theinstantaneous Epps e ff ect and notice that the Cuchiero-Teichmann estimates are not stable for di ff erent values of ∆ t for fixed value of time t . (a) MM Surface plot, N = Nyquist (b) MM Contour plot, N = Nyquist (c) MM, N = Nyquist, M = ff usion model with deterministic correlation in eqs. (4.1) and (4.2). We simulate n = , ff erent values for M when N is the Nyquistfrequency. Figure 5c compares the Malliavin-Mancino spot correlation (blue line with label “Estimated MM”) using M =
20 and N = Nyquist against the trueinstantaneous correlation (black line with label “True”). We see that for the very simple instantaneous correlation, larger M presents more zigzagging behaviourwhile a small M is adequate. inhomogeneity lead to di ff erent Epps curves. These di ff erentcurves require di ff erent time-scales to remove the decay. Thiscan be seen in Figures 6a to 6c. Therefore, the choice of N should depend on the specific Epps curve from the specific caseof asynchrony.A property of the Epps curves is that after the time-scale ∆ t has reached the saturation level, the correlations remain at thatlevel for even larger ∆ t . This allows us to find an appropriatetime-scale without needing knowledge of the “true” underlyingcorrelation levels such as the case of picking N to minimize theerror with respect to the MSE. Concretely, we can pick N basedon the minimum ∆ t which achieves the saturation level in theEpps curves. This allows the decay in correlation caused by theEpps e ff ect to be removed while providing the largest N possi-ble to provide a good approximation of eq. (2.3) using the Bohrconvolution product. It remains unclear how one should pickthe appropriate M after a suitable choice of N . However, as aconsequence of the sampling theorem, M can only be investi-gated for M ≤ N / M once N has beenchosen. This procedure is a purely pragmatic approach basedon the understanding of the Epps e ff ect to deal with the issue ofasynchrony.To demonstrate this pragmatic approach, let us consider asimple bivariate di ff usion defined as: dX it = σ i dW i ( t ) , i = , , (4.1)where the Brownian motions W and W have a deterministic Note that the Epps curves refer to the integrated correlation plotted as afunction of the sampling interval ∆ t . Note that for the Malliavin-Mancino estimates, larger ∆ t can present a de-viation from the saturation level because of small N which increases the vari-ability of the estimates. This can be seen in for example Ren`o (2003) when thecutting frequency is too small. instantaneous correlation given by: ρ ( t ) = sin ( t π T ) , (4.2)for t ∈ [0 , n = ,
800 gridpoints, T = σ , σ ) = (0 . , . ∆ t . Figure 5we establish the ground truth to see what choices of M recovera good approximation of eq. (4.2). Figures 5a and 5b visualisesthe Malliavin-Mancino spot correlation estimates for variousvalues of M with the Nyquist choice for N as the surface andcontour plots respectively. Figure 5c compares the Malliavin-Mancino (blue line) correlation spot estimates for M =
20 and N = Nyquist against the theoretical correlation (black line). Wesee that for the simple deterministic correlation, a small value of M is su ffi cient in approximating the spot correlation. Moreover,additional harmonics M presents fluctuations in the estimatesthrough the adding redundant frequencies.To demonstrate the need to pick N based on the Epps curves,three cases of asynchronous sampling is used. The synchronousgrid is sampled using an exponential inter-arrival with mean 10,20 and 50 for the three cases. Each case yielding n i ≈ n /λ for each asset. By plotting the integrated correlation as a func-tion of the time-scale ∆ t in Figures 6a to 6c, we see that dif-ferent levels of asynchrony reach the saturation level at di ff er-ent time-scales. Therefore, the time-scale required to amelio-rate the Epps e ff ect should be checked on a case-by-case basis.Here we use the Malliavin-Mancino integrated volatility / co-volatility estimates using the Dirichlet kernel to estimate theintegrated correlation. We use the Dirichlet kernel because itbetter recovers the theoretical Epps e ff ect arising from asyn-chrony (Chang et al., 2020b). The three time-scales (orangevertical line) which remove the Epps e ff ect while preservingthe largest N are ∆ t = ,
100 and 220 seconds. These choicesresult in N = ,
143 and 64 respectively. Figures 6d to 6f andFigures 6g to 6i are surface and contour plots for the Malliavin-Mancino spot estimates with M ranging from 1 to 119, 71 and32 respectively. We see that these choices of N allow us to re-cover correlations that are no longer around zero. However, the11
100 200 3000.10.20.30.4 (a) MM, 1 /λ = (b) MM, 1 /λ = (c) MM, 1 /λ =
20 40 60 80 100 0.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (d) MM, 1 /λ = ∆ t =
10 20 30 40 50 60 700.0 0.2 0.4 0.6 0.8-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (e) MM, , 1 /λ = ∆ t = (f) MM, 1 /λ = ∆ t = /λ = ∆ t =
60 (h) MM, , 1 /λ = ∆ t =
100 (i) MM, 1 /λ = ∆ t = (j) MM, 1 /λ = M = ∆ t = (k) MM, 1 /λ = M = ∆ t = (l) MM, 1 /λ = M = ∆ t = N on a case-by-case basis. The simple di ff usion model with deterministic correlation in eqs. (4.1) and (4.2) is simulatedfor n = ,
800 synchronous grid points. Three cases of asynchrony is then introduced by sampling each of the synchronous grid points with an exponential inter-arrival with mean 10, 20 and 50 for the three cases. These are columns one to three respectively. Figures 6a to 6c is the Malliavin-Mancino integrated correlationwith the Dirichlet kernel plotted as a function of the time-scale ∆ t . We pick N based on ∆ t which ameliorates the Epps e ff ect, resulting in ∆ t = ,
100 and 220 forthe three cases. Figures 6d to 6f and Figures 6g to 6i are surface and contour plots for the Malliavin-Mancino spot estimates with M ranging from 1 to 119, 71 and32 respectively. Figures 6j to 6l we compare the Malliavin-Mancino spot estimates (blue line with label “Estimated MM”) with M = ,
11 and 10 for the respectivethree cases against the true instantaneous correlation (black line with label “True”) from eq. (4.2). We see that in a simulation setting, the instantaneous correlationscan be recovered under the presence of asynchrony. M occur for much smaller valuesof M in the asynchronous case. The conundrum of picking anappropriate M is further exacerbated under asynchronous ob-servations. In the synchronous case, M =
100 presented fluctu-ations but they were relatively small in size. Here in the asyn-chronous case, massive fluctuations start appearing for M largerthan 20 even after choosing an appropriate N . It remains un-clear as to why this happens, but it is clear that under the pres-ence of asynchrony M needs to be smaller than the synchronouscase. This can be problematic if one is trying to approximatemore complex instantaneous dynamics such as the instanta-neous correlation from the Heston model as small M does notprovide enough harmonics for an accurate approximation, butlarger M is also not an appropriate option. Figures 6j to 6l wecompare the Malliavin-Mancino spot estimates (blue line) with M = ,
11 and 10 for the respective three cases against thetrue instantaneous correlation (black line) from eq. (4.2). Theseestimates are able recover the true correlations to a certain de-gree but are unsatisfactory compared to the synchronous case.This is only achievable under simulation conditions because weknow the ground truth so that the choice of M can be adjustedfor a satisfactory recovery. How one should pick M for the em-pirical data remains unclear, the only conditions we know isthat M must be small and must be less than N / M withempirical data, there is another issue with picking small N toremove the Epps e ff ect. By choosing large ∆ t to remove theEpps e ff ect hides the genuine e ff ects resulting in a decay ofcorrelations. This does not present an issue when the under-lying correlation does not depend on ∆ t such as di ff usion mod-els or when there is no lead-lag. However, when these genuinee ff ects are present the better approach is to disentangle statis-tical e ff ects causing the Epps e ff ect from the genuine e ff ects atvarious ∆ t . In the case of the integrated correlation, disentan-gling lead-lag from asynchrony has been done by Mastromatteoet al. (2011), and disentangling correlations that dependent on ∆ t from asynchrony has been done by Chang et al. (2020c). Asa preliminary investigation into the instantaneous Epps e ff ect,here we will only consider the impact from changing ∆ t ratherthan trying to disentangle the various e ff ects.
5. Empirical
Here we aim to demonstrate the instantaneous Epps ef-fect and ameliorate the e ff ect with empirical Trade and Quote(TAQ) data. To this end, we obtain two banking equities fromthe Johannesburg Stock Exchange (JSE) for the week from24 / / / / T = ,
200 seconds. Here we investigate each day sep-
Figure 7: Here we plot the integrated correlation estimates for the equity pairSBK / FSR at various time-scales ∆ t . The estimates are obtained using theMalliavin-Mancino integrated volatility / co-volatility estimate with the Dirich-let kernel. The time-scale is adjusted using eq. (3.6) and ranges from 1 to 400seconds. The Epps curves are plotted for each day of the week from 24 / / / / /
06 is the blue line, Tuesdaythe 25 /
06 is the orange line, Wednesday the 26 /
06 is the green line line, Thurs-day the 27 /
06 is the purple line, and Friday the 28 /
06 is the dark-green line. arately because the trading days are not connected which cana ff ect the spot estimates as they are built upon harmonics.We begin by plotting the integrated correlation as a functionof the sampling interval ∆ t for each day using the Malliavin-Mancino integrated volatility / co-volatility with the Dirichletkernel. The time-scale is adjusted using eq. (3.6) and rangesfrom 1 to 400 seconds. We see in Figure 7 that the integratedcorrelation between the days for the same equity pair can varyconsiderably, but they all exhibit the Epps e ff ect. To avoidexcessive figures, we will investigate the instantaneous corre-lation for Tuesday 25 / / / / ∆ t =
300 seconds and they present di ff erent saturationlevels.Next we demonstrate that the instantaneous Epps e ff ect ispresent with empirical data. The first and second row in Fig-ure 9 plots the surface and contour plots of the Malliavin-Mancino and Cuchiero-Teichmann for fixed M =
10 and vary-ing ∆ t respectively. The first and second columns are the sur-face and contour plot for Tuesday the 25 / / / / ∆ t is controlled using eq. (3.6) for the Malliavin-Mancinoestimator, while the Cuchiero-Teichmann estimator adjusts thetime-scale using the previous tick interpolation. First, we seethat the instantaneous Epps e ff ect is present, as in the spot es-timates are around zero for small ∆ t and increase when ∆ t in-creases. Second, as with the simulation experiments, the pre-vious tick interpolation presents instabilities in the Cuchiero-Teichmann estimates for fixed time t and varying ∆ t seen as To start the previous tick interpolation for each day, we set t = Figure 8: Here we compare the Malliavin-Mancino (solid lines) and Cuchiero-Teichmann (dashed lines) instantaneous estimates for M =
10 and ∆ t = / / / / the horizontal black marks in the contour plots. Third, the in-traday correlation structure can vary between the days for thesame equity pair. In the Malliavin-Mancino estimates, we seethat on Tuesday the 25 / / / / / / / / ∆ t = N =
46 and therefore the range of M to investigate is from 1 to 23. The first and second row ofFigure 10 plots the surface and contour plots of the Malliavin-Mancino and Cuchiero-Teichmann for fixed ∆ t =
300 and vary-ing M respectively. The first and second columns are the sur-face and contour plot for Tuesday the 25 / / / / M . What we do notice is that additional M buildsupon the harmonics which increases the fluctuation in the esti-mates but also allows the estimates to reach higher peaks andlower troughs.Without a clear method to pick M , we simply compare thetwo estimators for M =
10. Figure 8 compares the Malliavin-Mancino (solid lines) and Cuchiero-Teichmann (dashed lines)estimates for M =
10 and ∆ t = / / / / ∆ t meaning the dynamics obtained here can com-pletely change for similar ∆ t . Second, without some resolutionfor the instantaneous Epps e ff ect it remains unclear how one can examine the e ff ect of jumps in the estimators under the presenceof asynchrony.At this stage, the Malliavin-Mancino estimator is the pre-ferred estimator for ultra-high frequency finance. At this scale,the impact of asynchrony takes outweighs the impact of jumps.Therefore, the ability to bypass the time domain and avoidingthe need to impute data so that the estimates are more stablemakes the Malliavin-Mancino estimator more attractive.
6. Conclusion
In this paper, we compared the Malliavin-Mancino andCuchiero-Teichmann Fourier spot volatility estimators for var-ious Stochastic models with and without jumps. In the syn-chronous case, both estimators recover the entire volatility ma-trix with high fidelity when there are no jumps. With jumps,both estimates are able to recover the co-volatility but only theCuchiero-Teichmann faithfully recovers the volatility. In theasynchronous case and investigating small time-scales ∆ t = ff ect.We investigated the impact of various cutting frequencies M and N in the estimators. We find that the reconstruction fre-quency M plays a key role in the accuracy of the approxima-tion for the spot estimates and that the choice of M should de-pend on the composition of the true spot parameter of interest.The choice of M needs to achieve a subtle balance betweenachieving a good approximation without adding too many re-dundant frequencies. This problem is further exacerbated un-der the presence of asynchrony, where the large fluctuationsappears for smaller choices of M . There is currently no clearchoice for M under asynchrony, but the choice should be smalland must satisfy M ≤ N / ff ect arising fromasynchrony under simulation. The time-scales are controlledthrough N for the Malliavin-Mancino estimator while the pre-vious tick interpolation is applied for the Cuchiero-Teichmannestimator. We find that the Malliavin-Mancino estimates pro-duce stable estimates for di ff erent ∆ t because it bypasses theneed for interpolation in the time domain; while the Cuchiero-Teichmann estimates are unstable for varying ∆ t , because oftheir use of previous tick interpolation.We provide an ad hoc approach to deal with asynchrony bypicking N such that we have a time-scale that reaches the satu-ration level of the Epps curves. Although the approach is moreversatile than the choices of N presented in the literature, whichare given for specific cases of asynchrony; it still remains naive(as an approach to correct for the Epps e ff ect) because it canstill conceal genuine causes of the Epps e ff ect from statisticalones.Finally, we apply the estimators to Trade and Quote datafrom the Johannesburg Stock Exchange for two banking equi-ties. We demonstrate the instantaneous Epps e ff ect in the SouthAfrican equity market and once again find that the Malliavin-Mancino estimator obtains stable estimates for various ∆ t com-pared to the Cuchiero-Teichmann estimator. We find that the14
00 200 300 09:00 12:55 16:50-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (a) MM, M =
10, 25 /
06 (b) MM, M =
10, 25 /
100 200 300 09:00 12:55 16:50-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (c) MM, M =
10, 26 /
06 (d) MM, M =
10, 26 /
100 200 300 09:00 12:55 16:50-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (e) CT, M =
10, 25 /
06 (f) CT, M =
10, 25 /
100 200 300 09:00 12:55 16:50-1.0-0.50.00.51.0 -1.00-0.75-0.50-0.2500.250.500.751.00 (g) CT, M =
10, 26 /
06 (h) CT, M =
10, 26 / ff ect for empirical data. The first and second row plots the surface and contour plots of the Malliavin-Mancino and Cuchiero-Teichmann for fixed M =
10 and varying ∆ t respectively. The first and second columns are the surface and contour plot for Tuesday the25 / / / / ∆ t is controlled using eq. (3.6)for the Malliavin-Mancino estimator, while the Cuchiero-Teichmann estimator adjusts the time-scale using the previous tick interpolation. First, we see the Eppse ff ect is present for the instantaneous correlations; second, the Cuchiero-Teichmann presents instabilities in the estimates for varying ∆ t . (a) MM, ∆ t = /
06 (b) MM, ∆ t = / (c) MM, ∆ t = /
06 (d) MM, ∆ t = / (e) CT, ∆ t = /
06 (f) CT, ∆ t = / (g) CT, ∆ t = /
06 (h) CT, ∆ t = / M ranging from 1 to 23 after accounting for the Epps e ff ect by picking ∆ t = ∆ t =
300 and varying M respectively. The first and second columns arethe surface and contour plot for Tuesday the 25 / / / / M allows us to achieve higher peaks and lower troughs but at the cost of additional fluctuation. ff ect so that a correc-tion for asynchrony may be applied at various time-scales, suchas the correction by Chang et al. (2020c) and M¨unnix et al.(2011), but for the case of the instantaneous correlation. An-other potentially interesting area may be to apply the instan-taneous correlation estimates into clustering algorithms suchas the Agglomerative Super-Paramagnetic Clustering algorithm(Yelibi and Gebbie, 2019) to see if the variation in the intradaycorrelation dynamics result in di ff erent market states comparedto the static viewpoint traditionally used. Reproducing the Research
The data used in the research can be found in Chang et al.(2020a). The results can be reproduced by running the scriptfiles in the GitHub resource (Chang, 2020). Additional GIFs forthe surface and contour plots can also be found in the GitHubresource.
Acknowledgements
The author would like to thank Tim Gebbie and Etienne Pien-aar for various valuable conversations and critique. In particularI would like to thank Tim Gebbie for introducing me to this areaof research and the various related research problems, and formany detailed technical conversations and help with preparingthe paper. All remaining typographical and technical errors aremine.
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